[Paper Review] Are Newtonian Gravitation and Geometrized Newtonian Gravitation Theoretically Equivalent?

James Owen Weatherall|PhilSci-Archive (University of Pittsburgh)|Nov 21, 2014

Relativity and Gravitational TheoryPhysics and Astronomy27 references3 citations

TL;DR

This paper argues that Newtonian gravitation and geometrized Newtonian gravitation (Newton-Cartan theory) are theoretically equivalent under a revised criterion of theoretical equivalence, which better captures intuitive notions of physical equivalence than Clark Glymour's original criterion. The key result is that, on one standard interpretation of Newtonian gravity, the two theories are categorically equivalent, resolving a longstanding debate about their theoretical identity despite differing geometric formulations.

ABSTRACT

I argue that a criterion of theoretical equivalence due to Clark Glymour [Nous 11(3), 227-251 (1977)] does not capture an important sense in which two theories may be equivalent. I then motivate and state an alternative criterion that does capture the sense of equivalence I have in mind. The principal claim of the paper is that relative to this second criterion, the answer to the question posed in the title is "yes", at least on one natural understanding of Newtonian gravitation.

Motivation & Objective

• To challenge Clark Glymour's criterion of theoretical equivalence, which fails to capture a robust sense of equivalence between physical theories.
• To propose an alternative criterion—categorical equivalence—that better reflects when two theories are physically equivalent despite differing mathematical formulations.
• To demonstrate that Newtonian gravitation and Newton-Cartan theory are theoretically equivalent under this new criterion, under a standard interpretation of Newtonian gravity.
• To clarify the interpretive implications of theoretical equivalence for the metaphysics and ontology of spacetime theories.

Proposed method

• The paper introduces a refined criterion of theoretical equivalence based on category theory, specifically categorical equivalence between the categories of models of two physical theories.
• It applies this criterion to two formulations of electromagnetism, showing they are equivalent under the new standard—unlike under Glymour’s criterion—thereby validating the new criterion’s adequacy.
• The paper formulates Newtonian gravitation and Newton-Cartan theory as categories of models, with structures including spacetime manifolds, degenerate metrics, derivative operators, and matter fields.
• It proves that, under a standard interpretation of Newtonian gravity (as a force theory with a scalar potential), the category of models of Newtonian gravity is categorically equivalent to that of Newton-Cartan theory.
• The proof involves constructing natural isomorphisms between the model categories, preserving all physical structure and dynamics.
• It further establishes that gauge transformations in the electromagnetic analogy correspond to diffeomorphisms in the gravitational case, reinforcing the structural equivalence.

Experimental results

Research questions

• RQ1 Does Glymour’s criterion of theoretical equivalence adequately capture the intuitive sense in which two physical theories may be considered equivalent?
• RQ2 Are Newtonian gravitation and Newton-Cartan theory categorically equivalent under a revised criterion of theoretical equivalence?
• RQ3 Can two theories with different geometric structures—such as curved spacetime vs. force-based gravity—still be considered physically equivalent?
• RQ4 What is the correct criterion for theoretical equivalence that preserves the intuitive notion of physical equivalence in the face of mathematical differences?
• RQ5 Under what interpretations of Newtonian gravitation is it equivalent to geometrized Newtonian gravity?

Key findings

• The paper demonstrates that, under a revised criterion of categorical equivalence, Newtonian gravitation and Newton-Cartan theory are theoretically equivalent when Newtonian gravity is interpreted as a theory with a scalar gravitational potential.
• The alternative criterion successfully identifies equivalent formulations of electromagnetism—such as different gauge choices—as physically equivalent, unlike Glymour’s criterion.
• The model categories of Newtonian gravitation and Newton-Cartan theory are shown to be categorically isomorphic when Newtonian gravity is formulated with a potential satisfying Poisson’s equation.
• The proof relies on constructing natural isomorphisms between the categories of models, preserving all physical and geometric structure, including the role of the gravitational potential and derivative operators.
• The analysis shows that the apparent disagreement over spacetime curvature between the two theories is not a physical disagreement but a consequence of different mathematical formulations.
• The result implies that the choice between force-based and geometrized formulations of gravity may be physically neutral, depending on the interpretation of the underlying theory.